Norm on a certain quotient C$^*$-algebra

Question

Let $\{A_i\}_{i\in I}$ be an arbitrary family of C$^*$-algebras, we may define their product as $$\prod_{i\in I}A_i=\{(a_i);~\sup_{i\in I}\|a_i\|<\infty\}.$$ We can also define $$\sum_{i\in I}A_i=\{(a_i);~\text{For any }\epsilon>0\text{ there only finitely many }i\in I\text{ for which }\|a_i\|\geq \epsilon\},$$ which can be shown to be the closure of the collection of those $(a_i)$ in the product which have only finitely many non-zero entries. $\sum_I A_i$ is a closed (two sided) ideal in $\prod_I A_i$ and we can form the quotient $$Q:=\prod_{i\in I} A_i/\sum_{i\in I} A_i.$$ In the blue book "An Introduction to K-Theory for C$^*$-algebras" (Lemma 6.1.3) it is shown that when $I=\mathbb{N}$, the norm on $Q$ is given by $$\|[(a_n)]\|=\limsup_{n\rightarrow\infty}\|a_n\|,$$ where $[(a_n)]$ denotes the coset of $(a_i)$. My question is does this statement generalise? Say if $I$ is a directed set, then would it be true that $$\|[(a_i)]\|=\limsup_{i\in I}\|a_i\|?$$ Prahlad's answer to this question seems to say that yes, this is true, but I have only succeeded in making minimal progress on this. Mimicking the proof given in the blue book, I have shown that $\|[(a_i)]\geq \limsup_I\|a_i\|$, but I am failing to see how the reverse inequality can be derived.